Theorem iunin1

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in Enderton p. 30. Use uniiun to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015)

	Ref	Expression
Assertion	iunin1	|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Proof

Step	Hyp	Ref	Expression
1		iunin2	|- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
2		incom	|- ( C i^i B ) = ( B i^i C )
3	2	a1i	|- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iuneq2i	|- U_ x e. A ( C i^i B ) = U_ x e. A ( B i^i C )
5		incom	|- ( U_ x e. A C i^i B ) = ( B i^i U_ x e. A C )
6	1 4 5	3eqtr4i	|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )